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Section 2.10

2.10 Convected Coordinates In this section, the deformation and strain tensors described in §2.2-3 are now described using convected coordinates (see §1.16). Note that all the tensor relations expressed in symbolic notation already discussed, such as U  C , FˆN i  i n i, F  lF , are independent of coordinate system, and hold also for convected coordinates.

2.10.1

Convected Coordinates

Introduce the curvilinear coordinates i. The material coordinates can then be written as X  X( 1,  2 , 3 )

(2.10.1)

d X  dX i Ei  d i Gi ,

(2.10.2)

so X  X i E i and

where G i are the covariant base vectors in the reference configuration, with corresponding contravariant base vectors G i, Fig. 2.10.1, with

G i  G j   ij

(2.10.3) current configuration

reference configuration

G2

X

X 2 , x2

g2

G1

x

g1

E 2 , e2 3

X ,x

3

E 1 , e1

X 1 , x1

Figure 2.10.1: Curvilinear Coordinates The coordinate curves form a net in the undeformed configuration (over the surfaces of constant  i ). One says that the curvilinear coordinates are convected or embedded, that is, the coordinate curves are attached to material particles and deform with the body, so that

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Section 2.10

each material particle has the same values of the coordinates iin both the reference and current configurations. In the current configuration, the spatial coordinates can be expressed in terms of a new, “current”, set of curvilinear coordinates

x  x( 1 ,  2 ,  3 , t) ,

(2.10.4)

i with corresponding covariant base vectorsg i and contravariant base vectors g , with

d x  dxi ei  d i g i

,

(2.10.5)

Example Consider a motion whereby a cube of material, with sides of lengthL ,0 is transformed into a cylinder of radius R and height H , Fig. 2.10.2.

R H

L0 L0

Figure 2.10.2: a cube deformed into a cylinder A plane view of one quarter of the cube and cylinder are shown in Fig. 2.10.3.

x2

X2

P X

L0 x

X1

p

R

x1

Figure 2.10.3: a cube deformed into a cylinder

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Section 2.10

The motion and inverse motion are given by

2R x  L0 1

x  χ (X) ,

X  X   X  1 2

1 2

2 2

X1 X 2

x2 

2R L0

x3 

H 3 X L0

(basis: e i)

X   X  1 2

2 2

and

X  χ 1 ( x) ,

   

2 2 L0 x1  x 2 2R L x2 2 X 2  0 1 x1  x 2 2R x L X 3  0 x3 H

X1 

   

2

(basis: E i )

Introducing a set of convected coordinates, Fig. 2.10.4, the material and spatial coordinates are

X  X( 1,  2 , 3 ) ,

L  X 1   0  1  2R  L  X 2   0  1 tan  2  2R  X3 

L0 3  H

and (these are simply cylindrical coordinates)

x1  1 cos 2 x  x( 1 ,  2 , 3 ) ,

x 2  1 sin 2 x3  3

A typical material particle (denoted by p) is shown in Fig. 2.10.4. Note that the position vectors for p have the same  i values, since they represent the same material particle.

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Section 2.10

X2

  2

x2

 4

2

p

p

X1

2 x1

1 1 1  R Figure 2.10.4: curvilinear coordinate curves

■

2.10.2

The Deformation Gradient

With convected curvilinear coordinates, the deformation gradient is i

F  gi  G ,

(2.10.6)

which is consistent with





dx  d j g j  d j gi  G i G j  F dX

(2.10.7)

The deformation gradient F, the transpose F T and the inverses F 1 , F T , map the base vectors in one configuration onto the base vectors in the other configuration: FG i  g i

F  gi  G i F 1  G i  g i F T  g i  G i FT  Gi  g i



F 1g i  G i F TG i  g i

Deformation Gradient

(2.10.8)

F Tg i  G i

Thus the tensors F and F 1 map the covariant base vectors into each other, whereas the tensors F  T and F T map the contravariant base vectors into each other, as illustrated in Fig. 2.10.5.

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Section 2.10

contravariant basis

G

F T

2

FT

G2

g2

g2

G1

g1 G1

g1

F

covariant basis

F 1 Figure 2.10.5: the deformation gradient, its transpose and the inverses

Components of F

F has different components with respect to the different bases: F  Fij Gi  G j  F ij Gi  G j  Fi  j Gi  G j  Fji G i  G j  f ij g i  g j  f ij g i  g j  f i jg i  g j  f ij g i  g j

   f  g

   f  g

   f  g

   f  g

j

ij

i

F 1  F  1 ij Gi  G j  F  1 Gi  G j  F  1 i G i  G j  F 1 j G i  G j 1

i

ij

1 ij

 gj

i

1  j i

 gj

i

 1 i j

 gj

g j

i

  G  G  F  G  G  F  G  G  F  G  G   f  g  g   f  g  g   f  g g   f  g g

FT  F T

i

j

i

i

T

j

j

   f 

T j i

T ij

i

ij

T j i

T ij

ij

j

   f  g

i

j

T i j

i

j



ij



T i j

j

i



j

j

i



i

F  T  F  T ij G i  G j  F  T G i  G j  F  T i G i  G j  F T  j G i  G T

ij

gi  g j

T ij

i



 gj  f

T



j

i



gi  g j  f

T



i j

j

gi  g j (2.10.9)

 

The components of F with respect to the reference basesG i , G i are

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X m x m  i  j

Fij  Gi FG j  Gi  g j 

F ij  G iFG j  G jkG i  g k

(2.10.10)

Fi j  G i FG j  G jk G i  g k  i x m X m  j

F ij  Gi FG j  G i  g j 

and similarly for the components with respect to the current bases. Components of the Base Vectors in different Bases Now









gi  FGi  Fmj Gm  G j Gi  F mj G m  G j G i  F mjG m i j

 F mjG m i j

 Fmi G m

 FimG m

(2.10.11)

showing that some of the components of the deformation gradient can be viewed also as components of the base vectors. Similarly,

 

Gi  F  1 g i  f

 

m

1

mi

g m  f  1 i g m

(2.10.12)

For the contravariant base vectors, one has

  G  G G  F  G  G G  F  G   F  G   F  G  F  G

gi  F T Gi  F  T

mj

i

m

 T mj

m

j

T  j m

 T j m

i j

T  i m

 T mi

m

m

i

j

m

i j

(2.10.13)

m

and

 

Gi  F T g i  f

2.10.3

T mi

 

gm  f

T i m

gm

(2.10.14)

Reduction to Material and Spatial Coordinates

Material Coordinates Suppose that the material coordinates X i with Cartesian basis are used (rather than the convected coordinates with curvilinear basisG ), i Fig. 2.10.6. Then

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X j X j  E E j  Ei j  i X i ,  i j X i j i i G  E  E E X j X j Gi 

 i  X i,

x j x j  e ej j  i X i  i j X i j gi  e  e x j x j gi 

(2.10.15)

and x j e j  Ei  Grad x X i X i E i  e j  gradX  G i gi  Ei gi  j x

F  gi  Gi  gi  Ei  F

1

(2.10.16)

which are Eqns. 2.2.2, 2.2.4. Thus Grad x is the notation for F to be used when the material coordinates X i are used to describe the deformation. reference configuration

current configuration

E2

X2

g2

E1

X

g1

X1

X3

Figure 2.10.6: Material coordinates and deformed basis

Spatial Coordinates Similarly, when the spatial coordinates x i are to be used as independent variables, then

 i  x i,

X j X j E Ej  j x i  i , x i j  i j i G  E  E X j X j Gi 

x j x j e e j  ei  j x i  i  i j x i j gi  e  e  ei x j x j

gi 

(2.10.17)

and

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 xi e i  E j  Gradx X j X j F1  Gi  gi  G i  ei  E j  ei  gradX  xi F  gi  Gi  e i  G i 

(2.10.18)

The descriptions are illustrated in Fig. 2.10.7. Note that the base vectorsG , i gi are not the same in each of these cases (curvilinear, material and spatial). x2

X2

F  gi  G i

G2

g2

g1

G1 X1

x1

F

1

 Gi  g i x2

X2

g2

E2 E1

g1 X1

F

x e  E j  Grad x X j i

F 1 

X j j Ei  e  grad X x

i

x1

x2

X2

G2

G1

i

e2 e1

X1

x1

Figure 2.10.7: deformation described using different independent variables

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2.10.4

Strain Tensors

The Cauchy-Green tensors The right Cauchy-Green tensor C and the left Cauchy-Green tensor b are defined by Eqns. 2.2.10, 2.2.13,

C  FT F C 1  F  1F  T b  FF

T

1

T

     G  g  g  G   g G  G  C  G  G   g  G  G  g   G g  g  b g  g   g  G G  g   G g  g  b  g  g  G i  gi g j  G j  gij Gi  G j  Cij G i  G j i

j

j

i

b F F

i

j

i

j

i

i

ij

j

i

j

ij

i

1

 1 ij

ij

i

j

i

j

i

i

ij

(2.10.19)

j

1

j

j

j

ij

Thus the covariant components of the right Cauchy-Green tensor are the metric coefficients g ij , the covariant components of the identity tensor with respect to the convected bases in the current configuration, I  g  g ijg i  g j. It is possible to evaluate other components of C, e.g. C ij , and also its components with respect to the current basis through 2.10.14, but only the components C ij with respect to the reference basis will be used in the analysis. Similarly,

for b 1 , the components b 1ij with respect to the current configuration will be used.

The Stretch Now, analogous to 2.2.9, 2.2.12,

ds 2  dx  dx  dXCdX

(2.10.20)

dS 2  dX  dX  dxb 1 dx so that the stretches are, analogous to 2.2.17,

2 

ds 2 dX dX ˆ Cd X ˆ C   dX 2 dS dX dX

dS 2 dx 1 dx    dxˆb1 dxˆ b  2 ds 2 dx dx 1

 dXˆ iCij dXˆ

  dxˆ

 dxˆ b i

j

1

(2.10.21) j

ij

The Green-Lagrange and Euler-Almansi Tensors The Green-Lagrange strain tensor E and the Euler-Almansi strain tensor e are defined through 2.2.22, 2.2.24,

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ds2  dS 2 1  dX C  I dX  dXEdX 2 2 2 2 ds  dS 1  dx I  b 1 dx  dxedx 2 2





(2.10.22)

The components of E and e can be evaluated through (writing G I, the identity tensor expressed in terms of the base vectors in the reference configuration, and g  I , the identity tensor expressed in terms of the base vectors in the current configuration)





1 1 1  C G  gij Gi  G j  Gij Gi  G j   gij  Gij Gi  G j  Eij Gi  G j 2 2 2 1 1 1 e  g  b1  g ij gi  g j  Gij gi  g j  g ij  Gij g i  g j  eij g i  g j 2 2 2 (2.10.23)

E









Note that the components of E and e with respect to their bases are equal, Eij  eij (although this is not true regarding their other components, e.g. E ij  e ij ).

2.10.5

Intermediate Configurations

Stretch and Rotation Tensors

The polar decompositions F  RU  vR have been described in §2.2.5. The decompositions are illustrated in Fig. 2.10.8. In the material decomposition, the material is first stretched by U and then rotated by R. Let the base vectors in the associated intermediate configuration be gˆ i  . Similarly, in the spatial decomposition, the material is first rotated by R and then stretched by v. Let the base vectors in the associated intermediate configuration in this case be G i  . Then, analogous to Eqn. 2.10.8, {▲Problem 1} UG i  gˆ i

U  gˆ i  G i 1

U  G i  gˆ

U T  gˆ i  G i



U  TG i  gˆ i

U T  G i  gˆ i

U T gˆ i  G i

v  g i  Gˆ i ˆ i  gi v 1  G

ˆ i  gi vG v 1g  Gˆ

T

v  g  Gˆ i ˆ i g vT  G i

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U 1gˆ i  G i

i

i



288

i

i

ˆ i  gi v G v T g i  Gˆ i T

(2.10.24)

(2.10.25)

Kelly

Section 2.10

Note that U and v symmetric, U  U T, v  v T , so

U  gˆ i  G i  G i  gˆ i

UGi  gˆi , Ugˆ i  G i



U 1  G i  gˆ i  gˆ i  G i

v  g i  Gˆ i  Gˆ i  g i ˆ i  gi  gi  G ˆi v 1  G

U 1 gˆ i  Gi ,

U1 G i  gˆ i

ˆ  g , vg i  G ˆi vG i i ˆ i  gi ˆ i , v 1 G v 1 gi  G



(2.10.26)

(2.10.27)

Similarly, for the rotation tensor, with R orthogonal,R  1 R T, ˆ  Gi  G ˆiG R G i i i i T ˆ ˆ R  Gi  G  G  G i

ˆi RG i  Gˆ i , RG i  G T T R Gˆ i  G i , R Gˆ i  G i



R  g i  gˆ i  g i  gˆ i

Rgˆ i  g i,



R T  ˆg i  g i  gˆ i  g i

Rgˆ i  g i

R T g i  gˆ i, R Tg i  gˆ i

(2.10.28)

(2.10.29)

The above relations can be checked using Eqns. 2.10.8 and F RU, F  vR , v 1  RF 1 , etc.

R

Gˆ  i

v

G i 

g i

U

gˆ i 

R

Figure 2.10.8: the material and spatial polar decompositions Various relations between the base vectors can be derived, for example, ˆ  g   RG   Rgˆ   G R T Rgˆ  G  gˆ G i j i j i j i j

Solid Mechanics Part III

ˆ i g j  G ˆ i g  G j



 G i  gˆ j



 G i  gˆ j

ˆ i g j  G



 G i  gˆ j

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(2.10.30)

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Section 2.10

Deformation Gradient Relationship between Bases The various base vectors are related above through the stretch and rotation tensors. The intermediate bases are related directly through the deformation gradient. For example, from 2.10.26a, 2.10.28b, ˆ  FT G ˆ gˆ i  UG i  UR T G i i

(2.10.31)

In the same way, ˆ gˆ i  F T G i i 1 ˆ i gˆ  F G

(2.10.32)

ˆ i  F T gˆ i G ˆ i  Fgˆ i G

Tensor Components The stretch and rotation tensors can be decomposed along any of the bases. For U the most natural bases would be G i  and G i , for example, U U ij G i  G j , U ij  G i UG j  G i  gˆ j U  U ij Gi  G j , U

ij

 G iUG j  G imG j  ˆg m

U  U ij G i  G j , U ij  G i UG j  G i  gˆ U  U i j G i  G j , Uij  G i UG j  gˆ i G

(2.10.33)

j j

with U ij  U ji , Uij  U ji , Ui j  Uji , Ui j  U ij . One also has

ˆi G ˆj, v G ˆ vG ˆ G ˆ g v  v ij G ij i j i j ˆ i vG ˆ j  Gˆ im G ˆ j g v  v ij Gˆ i  Gˆ j , v ij  G m ˆi  G ˆ j , v ij  G ˆ i vG ˆ j G ˆ i  gj v  vi j G 

(2.10.34)

ˆ iG ˆ , v j  G ˆ vG ˆ j  g G ˆj v  v i j G j i i i with similar symmetry. Also,

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  U  U  U

   

U U U U

U 1  U  1 ij gˆ i  gˆ j , U1 U 1 U 1

1 ij

ˆgi  ˆgj ,

1 i j

gˆi  gˆ j ,

1  j i

gˆi  gˆ j ,

   

1

 gˆ i U  1gˆ j  G i  gˆ j

ij

1 ij

 ˆgi U1 ˆgj  gˆ im Gm  gˆ j

1 i j

 gˆ i U 1gˆ j  gˆ i  G j

1  j i

 gˆ i U 1gˆ j  G i  gˆ j

(2.10.35)

and

  v  v  v

   

v 1  v  1 ij gi  g j , v1 v 1 v 1

1 ij

gi  g j ,

1 i j

gi  g j ,

 1 j i

gi  g j ,

v v v v

   

1
...